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Ternary Composition Algebras II. Automorphism Groups and Subgroups

Ronald Shaw
Proceedings: Mathematical and Physical Sciences
Vol. 431, No. 1881 (Oct. 8, 1990), pp. 21-36
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/79936
Page Count: 16
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Ternary Composition Algebras II. Automorphism Groups and Subgroups
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Abstract

We study an eight-dimensional ternary composition algebra E = (E, $\langle \ \rangle,\{\,\}$) of signature (8, 0) or (4, 4). Such an algebra is associated with an involutary outer automorphism M of the Lie algebra so(E) of $\langle,\rangle $. The automorphisms and (in the (4, 4) case) counter-automorphisms of E are obtained in terms of the d = 7 spin group determined by M. Using M we derive a `principle of duplicity' for the ternary multiplication {} (related to the well-known principle of triality for the associated octonionic binary multiplication). However, rather than construct E out of the octonions as done (in effect) by previous authors, we give a four-dimensional `complex' construction of E. This non-octonionic view of E highlights certain (15-dimensional) d = 6 spin groups, rather than the customary (14-dimensional) octonionic automorphism groups. In the (4, 4) case we can choose to interpret `complex' in terms of the split complex numbers, and are thereby led to consider the subgroup chain [Note: See the image of page 21 for this formatted text] SO+ (4,4) $\supset $ Spin+ (3,4) $\supset $ SL (4;R) (≃ Spin+ (3,3)) instead of the chain SO+ (4,4) $\supset $ Spin+ (3,4) $\supset $ SU (2,2) (≃ Spin+ (2,4)).

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